
 There are canonical isomorphisms \(\Psi _{\overline{\mathcal {J}}_{g,n}}:\Sigma (\overline{\mathcal {J}}_{g,n})\xrightarrow{} J^{trop}_{g,n}\)  and \(\overline{\Psi }_{\overline{\mathcal {J}}_{g,n}}:\overline{\Sigma }(\overline{\mathcal {J}}_{g,n})\xrightarrow{} \overline{J}^{trop}_{g,n}\)  of, respectively, generalized cone complexes and generalized extended cone complexes.

 Using the above isomorphism \(\overline{\Psi }_{\overline{\mathcal {J}}_{g,n}}\) , the map \(\widetilde{\operatorname{trop}}^{an}_{\overline{\mathcal {J}}_{g,n}}:|\overline{\mathcal {J}}_{g,n}^{\beth }|\rightarrow \overline{J}^{trop}_{g,n} \)  coincides with the map \(\operatorname{trop}^{an}_{\overline{\mathcal {J}}_{g,n}}:|\overline{\mathcal {J}}_{g,n}^{\beth }|\rightarrow \overline{\Sigma }(\overline{\mathcal {J}}_{g,n})\) , i.e. \( \widetilde{\operatorname{trop}}^{an}_{\overline{\mathcal {J}}_{g,n}}=\overline{\Psi }_{\overline{\mathcal {J}}_{g,n}}\circ \operatorname{trop}_{\overline{\mathcal {J}}_{g,n}}\) .
In particular, \(\widetilde{\operatorname{trop}}_{\overline{\mathcal {J}}_{g,n}}^{an}\)  is continuous, surjective and proper.

 The diagram (REF ) coincides with the diagram (REF ), i.e. we have the following commutative diagram
\({\big | \overline{\mathcal {J}}_{g,n}^{\beth }\big |@(ur,ul)[rrrr]^{\widetilde{\operatorname{trop}}^{an}_{\overline{\mathcal {J}}_{g,n}}} [d]^{|\Phi ^{\beth }|} [rr]_{\operatorname{trop}^{an}_{\overline{\mathcal {J}}_{g,n}}} && \overline{\Sigma }(\overline{\mathcal {J}}_{g,n}) [d]^{\Phi ^{\Sigma }} [rr]^{\overline{\Psi }_{\overline{\mathcal {J}}_{g,n}}}_{\cong }&& \overline{J}^{trop}_{g,n}[d]^{\Phi ^{\operatorname{trop}}} \\\big | \overline{\mathcal {M}}_{g,n}^{\beth }\big |@(dr,dl)[rrrr]_{\widetilde{\operatorname{trop}}^{an}_{\overline{\mathcal {M}}_{g,n}}}[rr]^{\operatorname{trop}^{an}_{\overline{\mathcal {M}}_{g,n}}} && \overline{\Sigma }(\overline{\mathcal {M}}_{g,n}) [rr]^{\overline{\Psi }_{\overline{\mathcal {M}}_{g,n}}}_{\cong } && \overline{M}^{trop}_{g,n}\\}\) 
where \(\overline{\Psi }_{\overline{\mathcal {M}}_{g,n}}:\overline{\Sigma }(\overline{\mathcal {M}}_{g,n})\xrightarrow{} \overline{M}^{trop}_{g,n}\)  is the isomorphism of generalized extended cone complexes constructed in [1]}.

 The two maps
\({\big |\overline{\mathcal {J}}_{g,n}\big | && \big |\overline{\mathcal {J}}_{g,n}^{\beth }\big | [ll]_{\operatorname{red}_{\overline{\mathcal {J}}_{g,n}}} [rr]^{\widetilde{\operatorname{trop}}^{an}_{\overline{\mathcal {J}}_{g,n}}} && \overline{J}^{trop}_{g,n}}\) 
are compatible with the stratifications (REF ) and (REF ), i.e.
\((\widetilde{\operatorname{trop}}^{an}_{\overline{\mathcal {J}}_{g,n}})^{-1}\big (\overline{J}^{trop}_{(G,D)}\big )=(\operatorname{red}_{\overline{\mathcal {J}}_{g,n}})^{-1}\big (|J_{(G,D)}|\big )\) 
for every \((G,D)\in \operatorname{\mathcal {QD}iv}_{g,n}\) .
In particular, we have that

\((\widetilde{\operatorname{trop}}^{an}_{\overline{\mathcal {J}}_{g,n}})^{-1}\big (\overline{J}^{trop}_{g,n,d}\big )=\big |\overline{\mathcal {J}}_{g,n,d}^\beth \big |=(\operatorname{red}_{\overline{\mathcal {J}}_{g,n}})^{-1}\big (|\overline{\mathcal {J}}_{g,n,d}|\big )\)  for any \(d\in \mathbb {Z}\) .

\((\widetilde{\operatorname{trop}}^{an}_{\overline{\mathcal {J}}_{g,n}})^{-1}\big (\overline{J}_{g,n,(d)}^{trop, spl}\big )=\big |(\overline{\mathcal {J}}_{g,n,(d)}^{\operatorname{spl}})^\beth \big |=(\operatorname{red}_{\overline{\mathcal {J}}_{g,n}})^{-1}\big (|\overline{\mathcal {J}}_{g,n,(d)}^{\operatorname{spl}}|\big )\) .

\((\widetilde{\operatorname{trop}}^{an}_{\overline{\mathcal {J}}_{g,n}})^{-1}\big (\overline{J}^{trop}_{g,n}(\phi )\big )=\big |\overline{\mathcal {J}}_{g,n}(\phi )^\beth \big |=(\operatorname{red}_{\overline{\mathcal {J}}_{g,n}})^{-1}\big (|\overline{\mathcal {J}}_{g,n}(\phi )|\big )\)  for any universal stability condition \(\phi \in V_{g,n}\) .



